Method for designing generalized spirals, bends, jogs, and wiggles for railroad tracks and vehicle guideways

ABSTRACT

A prior method for designing transition curves for railroad tracks and other vehicle guideways begin with a choice of a roll function representing a functional form for variation of the track or guideway roll or cant angle as a function of distance and requires the curvature of the transition shape to keep the components of centripetal and gravitational acceleration in the plane of the track or guideway equal at each point along the shape and integrates the equation expressing that equality as part of a procedure for determining the resulting transition curve shape. That method is supplemented by a method of defining basic roll functions in terms of Gegenbauer orthogonal polynomials, including roll functions which generate simple spirals as well as more complex shapes (referred to as bends, jogs, and wiggles). Roll functions for the various shapes are defined as weighted sums of the basic roll functions and can generate transition curve shapes that have good dynamic characteristics and that are more general than the shapes that can be constructed using the prior method. A resulting generalized spiral can be used to compensate for inadequate offset when a spiral needs to be lengthened for operation at higher speed or to realign an existing spiral whose shape has become so different from its original design shape that restoration to that shape would be impractical.

BACKGROUND OF THE INVENTION

In the field of geometrical layout for railroad tracks the traditionalelements have been the straight line (with constant curvature equal tozero), the circular arc (with curvature that is constant but not zero),and the spiral (along whose length the curvature varies monotonically).When two sections of track that have different constant values ofcurvature would otherwise meet one another it is normal (with exceptionsin some special cases) for the two sections to be connected by a spiralwhose curvature and compass bearing at each end matches those of theadjacent section to which that end connects. Spirals have traditionallybeen conceived as geometrical shapes on the ground, and a number ofspecific shapes have been devised and applied during the past twocenturies.

A method for the design of railroad track spirals, and a number ofspecific shapes that can be obtained using this method, are described inInternational Application No. PCT/US01/41074 by Louis T. Klauder, Jr.,titled “Railroad Cur-e Transition Spiral Design Method Based on Controlof Vehicle Banking Motion” (hereafter referred to as the “KS_Method”).The KS_Method looks at a spiral not first of all as a shape but ratheras means of helping the guided vehicles change their roll or bank anglefrom the value appropriate for getting gravity to provide centripetalacceleration in one section to the value appropriate for that purpose ina following section whose centripetal acceleration is different.

The following introduces some terminology that is helpful for describingthe general field of track and guideway geometry, the KS_Method, and themethods of the current invention. Let the speed of travel be denoted asv, and denote compass bearing of the track as a function of distance salong the track by b(s). The curvature is defined as the firstderivative of the bearing with respect to distance, which is denoted asdb(s)/ds. The component of centripetal acceleration in the plane of thetrack will be provided by gravity if the equationv ² db(s)/ds cos(r(s))=g sin(r(s))  (1)is satisfied, where g is the acceleration of gravity, r(s) is thefunction that specifies the roll or bank angle of the track as afunction of distance s, and cos and sin are the common trigonometryfunctions. Hereafter, the forgoing equation is referred to as the“balance equation”, and v is referred to as the “balance speed”.

In the KS_Method for designing a spiral the first task is to choose afunctional form for r(s) within the length of the spiral. The subsequenttasks are: to integrate the balance equation to obtain the compassbearing b(s), to integrate cos(b(s)) and sin(b(s)) to obtainrespectively the x and y coordinates of points along the spiral, and toidentify the parameters of the function r(s) for which the resultingshape properly connects to the adjacent sections of constant curvaturetrack, incorporating the forgoing two stages of integration into aniterative search for that purpose if need be. A transition shapeconnects properly to an adjacent straight or circular arc section if theend of the shape has a point in common with the line or arc and if theshape has the same compass bearing and curvature as the line or arc atthe point in common. The most prominent parameter of r(s) is normallythe length of the spiral.

Approximations can be introduced to simplify equation (1) (the balanceequation) and the integrals of cos(b(s)) and sin(b(s)) to obtain x and yrespectively. The most common simplification replaces each cosinefunction by unity and each sine function by its argument expressed inradians. These simplifications will hereafter be referred tocollectively as the “small angle” approximation. If the roll function isa polynomial in s and this simplification is applied, then both stagesof integration called for in the KS_Method (and in the method of thecurrent invention) can often be done in closed form so that numericalintegration and iteration are not required. This simplification providesa good approximation to the extent that r(s) and b(s) are both ≦0.1throughout the transition. Even when these two angles do not stay thatsmall, this approximation, while not so good mathematically, may stillgive geometries that are effective in practice.

The method of the present invention takes advantage of the previouslyknown principle that the axis about which the roll of the track takesplace does not need to be located in the plane of the track but can beat a specified height, which height is also a parameter of the spiral.

The method of the present invention provides solutions for two existingproblems in the field of railroad track transition curve geometry. Oneproblem can arise when an existing route is being upgraded to allowoperation at higher speed. If for a particular curve the speed increaseis being provided for by increasing the superelevation (or banking) andwithout change of the radius of or path followed by the curve, then theoffset between the curve and a neighboring straight section will beunchanged and the length of a standard spiral connecting them will beunchanged. The offset is the shortest distance from a circular extensionof the curve to a straight extension of the straight section. It isgenerally necessary in such a case to find some way to lengthen thespiral. Examples of ways that traditional spirals and circular arcs havebeen used to address this problem in the past can be found in thearticle titled “Optimation of transition length increase” by HenrykBaluch, published in the October 1982 issue of Rail International.

The other problem occurs when maintenance work is being planned toadjust the alignment of an existing spiral whose shape has becomedeformed by passing trains. The problem is whether, and if so how, tomathematically define the shape to which the spiral should be restored.If a system is in place for measuring the location of the track relativeto local fixed monuments and the original shape was mathematicallydefined and the existing shape has not drifted very far from theoriginal shape, the spiral can be restored to the original design shape.When the forgoing conditions are not all met, the practice has normallybeen to “smooth” the alignment so that curvature measured along thecorrected alignment becomes close to some form of running average of thecurvature of the previous deformed alignment. Alignments created bysmoothing of that kind have generally not been described by mathematicalformulae. As a result, alignments have tended to drift over time.

SUMMARY OF THE INVENTION

The method of the present invention supplements and extends theKS_Method previously referred to by firstly introducing a group of newbasic transition geometry shapes that are distinct from spirals and thatare hereafter referred to as “bends”, “jogs”, and “wiggles”. These newshapes induce relatively little undesirable fluctuation in dynamicresponses of passing vehicles and are well suited to serve as transitioncurves in certain track situations. The basic shapes are characterizedby the net changes of several quantities over their lengths as follows.

Traversing a spiral from end to end there is a net change in curvature.There is usually also a net change in bearing, but that is not the casewhen a single spiral connects symmetrical reverse curves.

Traversing a bend from end to end there is a net change in bearing anglebut no net change in curvature. An example of a bend is illustrated inFIG. 1. A bend will be the best geometry for connecting two straightsections whose relative compass bearing difference is small.

Traversing from end to end a jog that is designed to provide atransition from one straight section to another straight section that isparallel thereto but offset therefrom, there is no net change in bearingangle or curvature. An example of a jog is illustrated in FIG. 2. A jogwill provide good geometry for connecting two straight sections that areparallel but offset by a modest amount. A jog may provide a goodgeometry for a high speed crossover.

Secondly, the present invention shows that the preferred roll functionsfor basic spirals, bends, and jogs can be conveniently expressed interms of the Gegenbauer orthogonal polynomials of orders 1, 2, and 3.Corresponding expressions with Gegenbauer polynomials of orders 4 andhigher are taken as definitions of the roll functions of wiggles of thecorresponding orders. In addition to generalization through inclusion ofhigher integer values of the Gegenbauer polynomial order (i.e., thelower index n), a second generalization is provided through thenon-integer upper index usually denoted by lowercase Greek alpha butwritten herein as (m+½). Instead of restricting m to be an integer ≧1,it is sufficient to let m be any real value ≧1 with the value 2 apopular choice.

Thirdly, the present invention introduces the method of starting with abasic shape such as a spiral, bend, jog, or Wiggle and then augmentingits roll function by adding thereto the roll functions of one or morehigher or lower order shapes, each with an adjustable coefficient, sothat the original shape becomes more flexible. This method isparticularly applicable for designing mathematically defined spiralswith good dynamic characteristics and with shapes that do not depart asfar as a basic spiral would from some existing deformed track spiral, asillustrated in FIG. 5. It is also very applicable when an existingspiral needs to be lengthened to allow for higher operating speed but itis preferred not to increase the offset for the spiral by moving thewhole curve. An augmented spiral configured for this purpose isillustrated in FIG. 4.

As another example, a combination of spiral and jog roll functions withthe jog function predominating can provide good geometry for atransition from one to the other of two adjacent and concentric circularalignments (e.g., a long curved section of double track) that is muchshorter than the simple improved spiral connecting the two concentricalignments. Within a combination of basic roll functions based onGegenbauer polynomials, different basic roll functions could haveGegenbauer polynomials with different values of m, but a common value ofm among all the constituent basic roll functions is expected to be morepopular.

Traversing a Wiggle of order 4 from end to end there is a small netchange in bearing angle, there is no net change in curvature, andextensions of the adjacent sections are congruent. FIG. 3 illustrates anorder 4 Wiggle calculated using an approximation one of whoseconsequences is that the net change in bearing angle of the Wigglebecomes zero. Generally, the roll function of a Wiggle is augmented byaddition of a small bend factor times the roll function of a bend, andthe bend factor is adjusted so that the augmented Wiggle has the desiredzero or non zero value of net change in bearing. An order 4 Wiggle canprovide good geometry when what is otherwise straight track needs tomake a small lateral excursion to avoid some obstacle. An order 5 Wigglecan provide a good geometry when otherwise straight track needs to yieldlaterally first to one direction and then to the other in order to avoidsuccessive obstacles on opposite sides of the track.

Fourthly, shapes similar to those described above can be obtained usingthe basic and augmented roll functions described above but introducingapproximations, such as those which have been explained previously, tosimplify equation (1) (the balance equation) and the integrals ofcos(b(s)) and sin(b(s)) to obtain respectively the x and y coordinatesalong the transition.

Fifthly, if for one of the basic or augmented roll functions describedabove each Gegenbauer polynomial that appears is replaced by the finitesum of terms by which it is defined, multiplications are carried out,and terms with common powers of distance along the roll function arecollected, the result will be a roll function that may seem unrelated toGegenbauer polynomials. If such a roll function were presented as asingle function rather than as a combination of the basic functionsdefined herein, it would still be equivalent to the roll functionsdisclosed herein.

THE DRAWINGS

FIGS. 1, 2, and 3 illustrate respectively a simple bend, a simple jog,and a simple Wiggle.

FIG. 4 illustrates a spiral augmented with a bend component.

FIG. 5 illustrates a spiral augmented by various combinations of higher.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In accordance with the present invention, methods are disclosed forconstructing new forms of roll functions that can be used in theKS_Method for constructing track and guide way curve transition shapes.With these new forms of roll functions the KS_Method can create spiralsthat are more flexible than the spiral shapes available heretofore, andit can also create transition shapes referred to as bends, jogs, andwiggles that have characteristics different from spirals and one fromanother, as previously described. Further, in accordance with thepresent invention, the “small angle” simplification method is disclosedfor designing bends, jogs, and wiggles.

In a first aspect of the present invention, for defining a set of basicroll functions, a basic roll function of integer order n is defined interms of its second derivative with respect to distance along the shapeby requiring the latter to be of the formk_(n)(a²−s²)^((m))C_(n) ^((m+1/2))(s/a), n>0  (2)where C_(n) ^(a)(x) is the standard Gegenbauer orthogonal polynomial asdefined in standard references (such as Abramowitz & Stegun, “Handbookof Mathematical Functions”, US Government Printing office, Washington,D.C., 1964, chapter 22), k_(n) is an adjustable constant, a is one halfthe length of the transition shape, s is distance along the shaperelative to the midpoint thereof, and m is a not necessarily integervalue ≧1. The value for m that is expected to be most useful is m=2.However, values such as m=1.5, 2.5, and 3 could also give usable shapes.It is not necessary that m be half integral, but when it is not C_(n)^((m+1/2))(s/a) will include non integral powers of x so that algebrawill be more complex.

The expressions for roll angle versus distance obtained by integratingequation (2) two times with respect to s takes the formj₁(integral on t from −a to s of (a²−t²)^((m+2)))  (3)when n=1, and the formj_(n)(a²−s²)^((m+2))C_(n−2) ^((m+5/2))(s/a), n>1  (4)when n>1, where j_(n) is a new constant coefficient. The integral ofequation (3) can be obtained in closed form when m is half integral. Forexample, for m=2 one finds that expression (3) for the roll angle versusdistance takes the formj₁(a+s)⁴(16a³−29a²s+20a s²−5s³)  (5)where j₁ has been redefined.

In a second aspect of the present invention, for forming roll functionsto be used in the KS_Method, basic roll functions of orders 1, 2, 3, 4,. . . can be used either by themselves or in linear combinations, wherethe term “linear combination” means the sum of a set of contributionseach of which has its own coefficient. A roll function that is a linearcombination of basic roll functions with a common value of m isidentified by an order symbol such as {m; 0.0, 1.0, 0.5} in which thecomma separated values following the semicolon indicate the values ofthe j_(n), coefficients for the basic shapes of orders 1, 2, 3, 4, . . .relative to the (normally unique) j_(n) that is =1.0. Several examplesof uses for linear combinations with more than one basic roll functionhave previously been described.

The basic roll functions of orders 2 and 3 generate bends and jogsrespectively. As previously explained, the order 4 roll function willgenerate a typical Wiggle by itself if the calculations are done aftermaking the “small angle” simplifications. When those simplifications arenot made, a Wiggle will typically need to have an order such as {2; 0,0.01, 0.0, 1.0} where the coefficient for the small bend component isadjusted so that the net change in compass bearing over the length ofthe Wiggle is zero.

When methods of this invention are being used to construct a rollfunction for a spiral that is augmented by an addition of bend and/orother higher order components, it is preferred to use the KS_Methodwithout application of the “small-angle” simplifications.

In a third aspect of the present invention the “small angle”simplifications are applied within the KS_Method so that the latterbecomes easier to use for the design of bends, jogs, and wiggles. Whenthe methods of this invention are being used to construct a rollfunction for a bend, a jog, or a Wiggle that is being fitted to apractical situation, compass bearing values will often be limited to arange small enough so that, with a suitable choice of axes, it may besatisfactory to apply the “small angle” simplifications within theKS_Method. When that is the case, integration of balance equation (1) toobtain b(s) and integration of the cosine and sine thereof to obtainrespectively the x and y coordinates along the shape can be carried outin closed form. The values that j_(n) and a must take in order for theshape to make the desired transition can then also be found in closedform so that the computational procedure is simplified. Application ofthe “small angle” simplification in this way is explained below for thecase of a bend. Differences that apply when working out analogousresults for a jog and a Wiggle are indicated. Details can be reproducedby one skilled in the art. When deriving the relevant formulae it ishelpful to use any one of a variety of known symbolic mathematicscomputer programs (such as Derive, Mathematica, or Maple). If thissimplified treatment is used in practice, one must be aware that therelationship between curvature and superelevation is slightly differentthan normal and may need to take account of that when choosing thebalance speed v to be used in the design.

For a simple bend with m=2, expression (4) givesr(s)=k(a ² −s ²)⁴  (6)where k is a constant to be determined. Working in the coordinate systemillustrated in FIG. 1, x, which is the integral of unity from zero to sbecomes simply s, meaning that s is not longer the distance along thebend but rather the x coordinate. With the “small angle” simplificationb(s) ceases to be the bearing angle and becomes instead the tangentthereof (hereafter written as bt(s) as a reminder). The integral of thesimplified form of balance equation (1) is found to givebt(x)=gkx(315a ⁸−420a ⁶ x ²+378a ⁴ x ⁴−180a ² x ⁶+35x ⁸)/(315v ²)  (7)

The y coordinate y(x) as a function of x is the integral of bt(x) from−a to x.

Carrying out the integration without regard to the constant ofintegration one obtainsy(x)=−gk(193a ¹⁰−315a ⁸ x ²+210a ⁶ x ⁴−126a ⁴ x ⁶+45a ² x ⁸−7x ¹⁰)/630v²)

With the lower limit of the forgoing integration at −a, y(x) of formula(8) will be zero at each end of the bend. It is necessary to add to thatresult the actual height of the two ends of the bend, namelya tan(turn/2)  (9)where turn denotes the bearing angle difference between the straightsections connected by the bend. The track is displaced downward from thepath of the roll axis by the overhang which ish sin(r(x))  (10)where h denotes the height of the roll axis above the plane of thetrack. Thus the formula for the y coordinate of the track isy _(—) track(x)=y(x)+a tan(turn/2)−h sin(r(x))  (11)

The primary constraint is that the bend turn by the correct amount. Thisconstraint has the formbt(x)=tan(turn/2)  (12)so that the constant k must be set tok=315v ² tan(turn/2)/(128a ⁹g)  (13)

There are two secondary constraints both of which place lower limits onthe value of the half length a. One is that the roll angle of the tracknot exceed the maximum allowed value denoted max_roll. The maximum rolloccurs at the center of the bend, and this constraint takes the forma_roll_lim=315v ² tan(turn/2)/(128g max_roll)  (14)where a_roll_lim is the first lower limit on a. The other secondaryconstraint is that the derivative of the roll angle with respect todistance not exceed the maximum allowed value denoted max_r_veloc. Themaximum value of dr(x)/dx occurs for x=−a/√7 and this constraints takesthe forma_twist_lim=9(308700)^(1/4)v (tan(turn/2))^(1/2)/(98(gmax_r_veloc)^(1/2))  (15)where a_twist_lim is the other lower limit on the half length.

In this simplified treatment the distance along the bend as a functionof x is obtained by numerical integration of the expression1/(cos(arctan(bt(x))−arctan(h cos(r(x))dr/dx)))  (16)and the actual length along the bend will be a little greater than 2 a.

When applying the “small-angle” simplification to the case of a jog or aWiggle the formulae for bt(x) and y(x) are obtained as above but basedon the formula for r(s) appropriate to the shape.

Looking at the case of a jog and using the coordinate system illustratedin FIG. 2, the lower limit in the integration to obtain y(x) isconveniently taken to be zero. The primary constraint is that thelateral displacement over the length of the jog, denoted jog_dist,should equal the specified distance between the parallel straightsections (or extensions thereof) to which the jog connects. As in thecase of the bend, the secondary constraints are that the roll angle andtwist of the track should nowhere exceed the respective limits chosenfor those two properties. The maximum values of roll angle and of rollvelocity occur at x=a/3 and x=0 respectively. The lower limits on thehalf length of the jog are found to bea _(—) roll _(—) lim=4(1155jog _(—) dist)^(1/2) v/(81(g max _(—)roll)^(1/2))  (17)anda _(—) twist _(—) lim=(6930jog _(—) dist v ²)^(1/3)/(8(g max _(—) r _(—)veloc)^(1/3))  (18)

In the application of the “small-angle” simplification to the case of aWiggle that makes an excursion to a distance swing_dist away from astraight line and then returns to that line, and using the coordinatesystem illustrated in FIG. 3, it is found that y(x) is proportional to(g/v²)(a²−x²)⁶. Consistent with the “small-angle” simplification thesine function is dropped from formula (10) above for the overhang. Thedistance from the straight line to the track is greatest at the centerof the Wiggle where that distance is y(0)+h r(0). The primary constraintis that the maximum excursion of the Wiggle from the straight line mustequal swing_dist. Applying that constraint determines the coefficient j₄of equation (4). Applying the secondary constraints one finds thata _(—) roll _(—) lim=2(3h max _(—) roll+3swing _(—) dist)^(1/2) v/(g max_(—) roll)^(1/2)  (19)anda _(—) twist _(—) lim=−4i(h/g)v sin(theta/3)  (20)where i is the square root of −1, andtheta=arcsin(i(hg)^(1/2) swing _(—) dist NC(h ² max _(—) r _(—) velocv))  (21)whereNC=(1517158400(3)^(1/2)/526153617+454246400/58461513)^(1/2)  (22)

The forgoing expressions for a_twist_lin are from solution of a cubicequation They can be evaluated easily using a known symbolic mathprogram such as Derive.

1. A method for constructing a roll function for use in designingtransition curves for railroad tracks and other vehicle guideways,wherein the designing of the transition curves requires the rollfunction to be supplied and wherein the method comprises the steps of:defining a set of basic roll functions; and constructing the rollfunction as a linear combination of at least one of the basic rollfunctions while treating coefficients of an individual basic rollfunction as parameters of the roll function and considering theindividual basic roll function to include a coefficient when theindividual basic roll function is referred to without mention of thecoefficient.
 2. A method according to claim 1 wherein the roll functionis used in a KS_Method for designing a transition shape and wherein themethod further includes the steps of: choosing a basic roll functionwhich specifies a variation of guideway roll angle as a function ofdistance and of adjustable parameters; causing centripetal andgravitational acceleration components in a plane defined by the guidewayto be equal at each point along a transition between two adjacentsections of the guideway by requiring curvature of alignment to satisfya balance equation; determining a resulting transition curve alignmentfor given values of adjustable parameters by integrating the balanceequation to obtain a compass bearing of the transition shape as afunction of distance and by then integrating the cosine and the sine ofthe compass bearing to obtain respectively x and y coordinates of pointsalong the transition shape, thereby defining a computed shape;determining parameter values for which the computed shape connects withthe two adjacent sections of the guideway; and repeating integrations ineach iteration of an iterative search.
 3. A method according to claim 2wherein the causing of the equal centripetal and gravitationalacceleration components to be equal and the determining of the resultingtransition curve alignment for given values of adjustable parametersfurther includes a small angle simplification including the steps ofreplacing a cosine function by unity and replacing a sine function by anargument of the sine function in radians if the argument of the cosinefunction or the sine function is the roll angle or the compass bearing,so that the curvature of the shape is defined directly as the rollfunction times a position independent factor.
 4. A method according toany one of claims 1 to 3 which further includes the step of defining thebasic roll functions via second derivatives of roll angle with respectto distance and in terms of standard Gegenbauer orthogonal polynomialsC_(n) ^(a)(x) by the formulad ² r(s)/ds ² =j _(n)(a ² −s ²)^(m) C _(n) ^((m+1/2))(s/a) where n is aninteger ≧1, m is a real value ≧1.0, a is one half the length of thetransition, s is a distance along the transition measured relative to amidpoint of the transition, r(s) is the roll angle as a function ofdistance s, and j_(n) is a constant, and wherein the basic rollfunctions are not defined as a linear combination of a single basic rollfunction where n=1.
 5. A method according to claim 4 wherein m is a realvalue selected from the group of values consisting essentially of 1.5,2, 2.5 and
 3. 6. A method according to any one of claims 1 to 5 fordesigning a generalized spiral transition and further comprising thestep of choosing a linear combination of the basic roll functions thatincludes more than one basic roll function and so that a net change inroll angle over the length of the transition is non zero.
 7. A methodaccording to claim 6 which further includes the step of adjusting theparameters of the generalized spiral so that the spiral connects from astraight section of the guideway to a curved section of the guideway,and after leaving the straight section, first moves away from the curvedsection and then reverses curvature to join the curved section, wherebythe generalized spiral can be made longer than a traditional spiralwithout being restricted by a lack of adequate offset betweenneighboring guideway sections.
 8. A method according to claim 6 whichfurther includes the step of adjusting the parameters of the generalizedspiral so that the spiral connects from one section of the guideway toanother section of the guideway and so that compared to a correspondingsimple spiral the shape of the generalized spiral lies closer to anexisting guideway transition having an alignment requiring improvement.9. A method according to claim 6 which further includes the step ofadjusting the parameters of the generalized spiral so that thegeneralized spiral connects from one section of the guideway to anothersection of the guideway and so that the generalized spiral is shaped toavoid a local obstruction.
 10. A method according to any one of claims 1to 5 for designing a bend transition and further comprising the stepsof: choosing a linear combination of the basic roll functions thatincludes at least one of the basic roll functions and so that a netchange in roll angle over the length of the transition is zero; andchoosing the basic roll functions so that the bend provides a transitionbetween two sections of the guideway which are both straight and notparallel with each other.
 11. A method according to any one of claims 1to 5 for designing a bend transition and further comprising the stepsof: choosing a linear combination of the basic roll functions thatincludes at least one of the basic roll functions and so that a netchange in roll angle over the length of the transition is zero; andchoosing the basic roll functions so that the bend provides a transitionbetween two sections of the guideway which are both circular arcs ofidentical radius with distinct centers and so that a line throughcenters of the two sections of the guideway is parallel to a linethrough two ends of the bend.
 12. A method according to any one ofclaims 1 to 5 for designing a jog transition and further comprising thesteps of: choosing a linear combination of the basic roll functions thatincludes at least one of the basic roll functions and so that a netchange in roll angle over the length of the transition is zero; andchoosing the basic roll functions so that the jog provides a transitionbetween two sections of the guideway which are both straight andparallel but not collinear.
 13. A method according to claim 12 whichfurther includes the step of adjusting parameters of the jog so that thejog defines a shape of at least a majority of a length of a crossoverbetween two sections of the guideway that run side-by-side in a twotrack configuration and that are both straight and parallel.
 14. Amethod according to any one of claims 1 to 5 for designing a jogtransition and further comprising the steps of: choosing a linearcombination of the basic roll functions that includes at least one ofthe basic roll functions and so that a net change in roll angle over thelength of the transition is substantially zero; and choosing the basicroll functions so that the jog provides a transition between twosections of the guideway which are both circular arcs of substantiallyidentical radius and that are substantially concentric.
 15. A methodaccording to claim 14 which further includes the step of adjustingparameters of the jog so that the jog defines a shape of at least amajority of a length of a crossover between two sections of the guidewaythat run side-by-side in a two track configuration and that are bothcircular arcs with radii that are substantially equal.
 16. A methodaccording to any one of claims 1 to 5 for designing a wiggle transitionand further comprising the steps of: choosing a linear combination ofthe basic roll functions that includes at least one of the basic rollfunctions and so that a net change in roll angle over the length of thetransition is zero; and choosing the basic roll functions so that if oneend of a resulting transition alignment connects to a particularstraight line, then another end of the resulting transition alignmentconnects to a location on the same straight line, and so that the wiggleenables an otherwise straight section to circumvent a local obstacle.17. A method according to any one of claims 1 to 5 for designing awiggle transition and further comprising the steps of: choosing a linearcombination of the basic roll functions that includes at least one ofthe basic roll functions and so that a net change in roll angle over thelength of the transition is zero; and choosing the basic roll functionsso that if one end of a resulting transition alignment connects to aparticular arc, then another end of the resulting transition alignmentconnects to a location on the same arc, and so that the wiggle enablesan otherwise uniformly curved section to circumvent a local obstacle.